## Note on the Chebyshev polynomials and applications to the Fibonacci numbers.(English)Zbl 0844.11012

Let $$(T_n (x))_{n\geq 0}$$ be the sequence of Chebyshev polynomials of the first kind defined by $$T_{n+2} (x)= 2x T_{n+1} (x)- T_n (x)$$ with $$T_0 (x)=1$$ and $$T_1 (x)=x$$. For these polynomials it is shown that the product of any two distinct elements of the set $$\{T_n, T_{n+2r}, T_{n+4r}, 4T_{n+r} T_{n+2r} T_{n+3r}\}$$, increased by $$({1\over 2} (T_h- T_k))^t$$, where $$t$$ ($$=1$$ or 2) and $$k> h\geq 0$$ are suitable integers, is a perfect square for any natural numbers $$n$$ and $$r$$. Applications and generalizations of the result are also shown.
Reviewer: Péter Kiss (Eger)

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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